University of Alberta Guide/STAT/222/Combining Continuous Random Variables

ConvolutionEdit

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ExampleEdit

 

  • Start by converting the pdf's to indicator functions
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      • Now   is defined only when   and   is defined only when  
  • Use the convolution formula above to write out the integral
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  • Factor out any constants, in this case, a multiplier
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  • Factor out the indicator function for   into the integral bounds
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      • Note that  
  • Now that have isolated the indicator for z, we can combine the entire integral for that indicator
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  • Finally, split the integral into the separate cases based on the remaining indicator function
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      • When   the integral has no bounds since   so the upper bound would be less than   which would be  .
      • When   the integral is bound between   and  since   will be at least   but less than  
      • As you can see there is a pattern here, it goes as follows:
        • Given   you will have